CAREER: Computation, Combinatorics, and Reality in Algebraic Geometry, with Applications

Sottile将致力于从代数几何和代数组合学的研究，到加强代数几何和应用科学之间的联系，再到教育推广等项目。 他的工作在代数几何将集中在具体方面的旗帜流形和他们的舒伯特品种，涉及一系列问题，从真实的舒伯特演算，从扩展量子舒伯特演算，从组合问题的舒伯特演算，并从几何舒伯特品种。 这将加深我们对计算，组合学和代数中的实数现象的理解。 几何 Sottile将研究结构特性的某些组合霍普夫代数，工作有应用多面体，舒伯特演算，并以某些组合 物理学中的方案。 他积极参与应用 计算代数 几何形状以 其他领域，特别是离散几何和应用科学。 在这一科学工作的一个统一的主题是研究具体的数学对象，使用计算机实验，以通知理论工作。虽然代数几何关注的是抽象的问题，方程的解决方案，它的价值，其他学科主要是通过具体的对象和计算工具。 该项目的一项主要活动是通过研究合作、组织会议和为应用科学家共同编写代数几何教科书，帮助将现代计算工具从代数几何传播到其他领域，包括应用科学。 Sottile还将为马萨诸塞州西部有才华的高中生组织一个外展计划，其特别目标是满足附近城市地区学生的需求，其中许多人来自传统上在科学方面代表性不足的群体。 这项工作的目的是：研究，技术转让，服务和推广，为Sottile的终身综合研究和教育活动提供坚实的基础。

Sottile will work on projects ranging from research in algebraic geometry and algebraic combinatorics, to strengthening links between algebraicgeometry and applied science, to educational outreach. His work in algebraic geometry will focus on concrete aspects of flag manifolds and their Schubertvarieties, involving a range of problems from the real Schubert calculus, from extensions of quantum Schubert calculus, from combinatorial problems in Schubert calculus, and from the geometry of Schubert varieties. This will deepen our understanding of computation, combinatorics, and real-number phenomena in algebraic geometry. Sottile will study structural properties ofcertain combinatorial Hopf algebras, work that has applications to polytopes, to the Schubert calculus, and to certain combinatorial schemes in physics. He is actively engaged in applications of computational algebraic geometry to other fields,particularly discrete geometry and the applied sciences. A unifying theme in this scientific work is studying concrete mathematical objects using computer experimentation to inform theoretical work.While algebraic geometry is concerned with abstract questions about solutions to equations, its value to other disciplines is largely through concrete objects and computational tools. A major activity of this project is to help disseminate the modern computational tools from algebraic geometry to other fields, including the applied sciences, through research collaboration, organizing conferences, and co-authoring a textbook on algebraic geometry for applied scientists. Sottile will also organize an outreach program for talented high school students in Western Massachusetts with the particular goal ofserving the needs of students in nearby urban areas, many of whom comefrom groups traditionally underrepresented in science. The intention of this work: the research, the technology transfer, the service and the outreach, is to provide a firm foundation for a lifetime of integrated research and educational activities for Sottile.